Graphs from Rings

Description

Preface Notation Contents Chapter 1 Introduction 1.1 Basics in Algebra 1.2 Basics in Graph Theory Chapter 2 Distances in Zero-divisor Graphs 2.1 Zero-divisor Graphs 2.2 Basic Properties of Zero-divisor Graphs 2.3 Girth of Zero-divisor Graphs 2.3.1 Rings of Polynomials and Power Series 2.3.2 Idealization 2.3.3 Rings with Linearly Ordered Primes 2.4 Diameter of Zero-divisor Graphs 2.4.1 Rings of Polynomials and Power Series 2.4.2 Idealization 2.4.3 Rings with Linearly Ordered Primes 2.5 Radius and Center of Zero-divisor Graphs 2.6 Rings without Identity Chapter 3 Properties of Zero-divisor Graphs 3.1 Coloring of Zero-divisor Graphs 3.1.1 Vertex Coloring 3.1.2 Clique Number 3.1.3 Edge Coloring 3.2 Connectivity of Zero-divisor Graphs 3.3 Isomorphisms and Automorphisms 3.4 Bipartite Zero-divisor Graphs 3.5 Zero-divisor Graphs of Gaussian Integers 3.6 Cycles in Zero-divisor Graphs 3.6.1 Trees and Complements 3.6.2 Eulerian and Hamiltonian Graphs Chapter 4 Genus of Zero-divisor Graphs 4.1 Genus of Graphs 4.2 Planar Zero-divisor Graphs 4.3 Zero-divisor Graphs with Genus One 4.4 Zero-divisor Graphs with Genus Two 4.5 Zero-divisor Graphs with Large Genus 4.6 Crosscap of Zero-divisor Graphs 4.7 Embedding of Line Graphs Chapter 5 Generalized Zero-divisor Graphs 5.1 Ideal-based Zero-divisor Graphs 5.2 Properties of ΓI(R) 5.2.1 Clique Number of Ideal-based Zero-divisor Graphs 5.3 Genus of Ideal-based Zero-divisor Graphs 5.4 Primal Ideal-based Zero-divisor Graphs 5.5 Annihilating-ideal Graphs 5.6 Special Annihilating-ideal Graphs 5.7 Coloring of Annihilating-ideal Graphs 5.8 Genus of Annihilating-ideal Graphs 5.9 Compressed Zero-divisor Graphs 5.9.1 Properties of Compressed Zero-divisor Graphs 5.9.2 Girth of Compressed Zero-divisor Graphs 5.10 Extended Zero-divisor Graphs Chapter 6 Zero-divisor Graph Generalizations 6.1 Noncommutative Rings 6.1.1 Directed Zero-divisor Graphs of Finite Rings 6.2 Commutative Semigroups 6.3 Commutative Semirings 6.4 Modules 6.4.1 Torsion Graphs of Modules 6.4.2 Zero-divisor Graphs of Modules 6.4.3 Annihilator Ideal-based Graphs of Modules 6.5 Domination in Directed Zero-divisor Graphs Chapter 7 Total Graphs of Commutative Rings 7.1 Definition of Total Graph 7.2 Basic Properties of Total Graphs 7.3 Polynomial Rings and Idealization 7.4 Properties of Total Graphs 7.4.1 Eulerian and Hamiltonian Graphs 7.4.2 Colorings 7.4.3 Connectivity 7.4.4 Domination 7.4.5 Intersection Graphs of Dominating Sets 7.5 Genus of Total Graphs Chapter 8 Graphs from Total Graphs 8.1 Total Graphs without the Zero Element 8.1.1 Basic Properties 8.1.2 Zero-divisor Paths and Regular Paths 8.2 The Regular Graph 8.3 Complements of Total Graphs 8.4 Generalized Complements of Total Graphs 8.5 Line Graphs of Total Graphs Chapter 9 Generalized Total Graphs 9.1 Commutative Ring Generalizations 9.1.1 Ideal-based Generalized Total Graphs 9.1.2 Total Graphs of Multiplicatively Closed Sets 9.1.3 Total Graphs of Multiplicative-prime Sets 9.2 Total Graphs of Modules 9.2.1 Total Torsion Element Graphs 9.2.2 Total Graphs of Modules 9.3 Commutative Semirings 9.3.1 Total Graphs of Semirings 9.3.2 Identity Summand Graphs Chapter 10 Other Graphs Associated with Rings 10.1 Graphs of Commutative Rings 10.1.1 Cayley Graphs 10.1.2 Co-maximal and Unit Graphs 10.1.3 Cozero-divisor Graphs 10.1.4 Jacobson and Intersection Graphs 10.2 Annihilator Graphs 10.2.1 Equal Annihilator and Zero-divisor Graphs 10.3 Compressed Annihilator Graphs 10.4 ΓI(R), ΓI(R), and AGI(R) 10.5 Congruence-based Graphs 10.6 Element Graphs 10.7 Dot Product Graphs 10.7.1 Subgraphs of Dot Product Graphs 10.7.2 Equivalence Dot Product Graphs 10.8 Trace Graph of Matrices 10.8.1 Basic Properties of Trace Graphs 10.8.2 Planarity and Thickness of Trace Graphs 10.8.3 Domination in Trace Graphs of Matrices 10.8.4 Ideal-based Trace Graphs of Matrices 10.9 Graphs of Noncommutative Rings 10.9.1 Commuting and Non-commuting Graphs 10.9.2 Nilpotent and Idempotent Graphs 10.10 Graphs of Ideals 10.11 Divisor Graphs 10.12 Subgroup Graphs of a Group Bibliography Index